Screen Time helps parents limit access to features and restrict usage to periods of time in iOS 13 and iPadOS 13—as well as monitor one’s own usage. In iOS 13.3 and iPadOS 13.3, Apple extended parental controls to whom a kid can be in contact with. (You can also use it to set your own communications preferences, but can override it as you wish.)

A parent can lock communications during downtime to specified contacts.
In the new view, Settings > Screen Time > Communication Limits, the person with the Screen Time passcode can control who contacts the device (or synced devices) or is contacted by the device’s user via iPhone, FaceTime, and Messages relying in large part on entries in the Contacts list.
iCloud syncing of Contacts must be enabled for this feature to work, which also effectively locks access to Contacts so that kids can’t modify entries or add new ones. (At this writing, at least two bypasses have been discovered that Apple will certainly patch. One allows an unknown number sending a text message to have its number added to contacts from the warning screen that blocks the message. The other allows Siri on an Apple Watch to call or text any number.)
There are two sections for communications: one is while access to a device is permitted during allowed screen time, and the other when general access is denied outside of exceptions set for particular apps. Even if Screen Time allows phone calls, FaceTime, or Messages during downtime, the limits set here take precedence on with whom.

A contact available during downtime is shown with a special label and some additional information.
In During Allowed Screen Time, communication can be set to Everyone or Contacts Only. An extra switch allows group messages in which the child’s account is part to add people who aren’t in Contact so long as an iCloud contact or someone in their Family Sharing list is part of the group.
The During Downtime section is even more restrictive, offering an option to allow communication only with Specific Contacts, who must be added via a Contacts browser or editor.
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